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Full text of "New circularly polarized frequency independent antennas with conical beam or omnidirectional patterns"

'^^/ ; 




LIB R.AFLY 

OF THE 

U N IVER.SITY 

OF ILLI NOIS 

621.365 

I^G55te 

no. 40-49 

cop. 2 



Digitized by the Internet Archive 
in 2013 



http://archive.org/details/newcircularlypol46dyso 



ANTENNA LABORATORY 



Technical Report No. 46 



NEW CIRCULARLY POLARIZED FREQUENCY 
INDEPENDENT ANTENNAS WITH CONICAL BEAM 
OR OMNIDIRECTIONAL PATTERNS 



by 

John D. Dyson and Paul E. Mayes 
20 June 1960 



Contract AF33(616)-6079 
Project No. 9-(13-6278) Task 40572 



Sponsored by: 
WRIGHT AIR DEVELOPMENT CENTER 



Electrical Engineering Research Laboratory 

Engineering Experiment Station 

University of Illinois 



Urbana, Illinois 



This report was presented, in condensed form, as a paper at the 
URSI-IRE Spring Meeting, Washington, D.C., 3 May 1960, under the 
title, "The Log-Spiral Omnidirectional Circularly Polarized 
Antenna." 



i i 3 \pZ> 

- ENGINEERING LIBRARY 






ACKNOWLEDGMENT 
The authors are pleased to acknowledge the assistance of Professor 
G. A. Deschamps in formulating the theory of excitation of multi-arm 
antennas. Discussions with W. T. Patton were helpful. 0. L. McClelland 
supervised the measurements program. 



ABSTRACT 

A conical beam may be obtained from balanced equiangular spiral 
antennas by constructing an antenna with more than two spiral arms and 
symmetrically connecting these arms to provide a suppression of the 
radiated fields on the axis of the antenna. The angle of this conical 
beam can be controlled and with proper choice of parameters it can be 
confined to the immediate vicinity of the azimuthal (0 = 90 ) plane. 

An antenna with four symmetrically spaced arms can provide a 
radiation pattern that is within 3 db of omnidirectional circularly 
polarized coverage. The standing wave ratio of this antenna referred 
to a 50 ohm coaxial cable is less than 2 to 1 over the pattern bandwidth, 

This four-arm version retains the wide frequency bandwidths of the 
basic conical log-spiral antenna, and it provides a coverage which here- 
tofore has been difficult to obtain even with narrow band antennas. 



CONTENTS 

Page 

1. Introduction 1 

2. The Conical Log-Spiral Antenna 3 

2.1 The Basic Structure 3 

2.2 The Radiation Pattern Beamwidth 5 

3. The Conical Beam Antenna 7 

3.1 The Principle 7 

3.2 Radiation Patterns 9 

3.3 Pattern Beamwidth 17 

3.4 The Input Impedance 17 

3.5 Operating Bandwidth as a Function of Antenna Size 17 

4. A Non-Frequency Independent Version 20 

5. Conclusions 24 
Appendix 25 



ILLUSTRATIONS 

Figure 

Number Page 

1. A conical log-spiral antenna with associated coordinate system 4 

2. Variation in electric field pattern of typical balanced 2 arm conical 
equiangular spiral antenna 6 

3. Possible feeding arrangements for multi-arm structures 8 

4. "infinite balun" feed used on a four arm conical beam antenna 10 

5. Typical electric field patterns and orientation of the conical beam 

as a function of the rate of spiral 11 

6. Antenna C - 15 - 9 etched from copper clad teflon impregnated 
fiberglass 13 

7. Electric field patterns of a balanced symmetrical 4 arm conical 
equiangular spiral antenna 14 

8. Electric field patterns of a balanced symmetrical 4 arm conical 
equiangular spiral antenna 15 

9. Azimuthal coverage of the radiation patterns in Figures 7 & 8 16 

10. VSWR of typical 4 arm conical equiangular spiral 18 

11. Projection of equiangular spiral and Archimedes spiral curves on 

a conical surface 21 

12. Electric field patterns of symmetrical 4 arm conical antennas 23 

1A. Terminal region of a structure having N-fold rotational symmetry 26 

2A. Instataneous electric vectors at 9 = and 6=9 for three values 

of m ° 30 



1. INTRODUCTION 

The balanced planar and conical equiangular spiral antennas have been 
demonstrated to have essentially frequency independent radiation and impedance 

characteristics over bandwidths which are at the discretion of the design 

12 * 

engineer ' . These antennas, based upon the equiangular or log-spiral curve, 

have the property that the highest and lowest usable frequencies are independent 

The highest usable frequency is determined by the diameter of the truncated 

region at the origin, which must remain small in terms of the operating 

wavelength, and the lowest usable frequency by the arm length and hence 

the maximum diameter of the antenna. 

The two-arm planar antenna provides circularly polarized, single lobe, 
bidirectional radiation on the axis of the antenna. An orthogonal projection 
of the two-arm planar antenna on a conical surface forms an antenna which, 
over a suitable range of parameters, confines the radiation to a single 
lobe directed off the apex of the cone. 

It is possible to devise a multitude of frequency independent structures 
by using the log-spiral arm as a basic element. Many different excitations 
may be used when several log-spiral arms with a common origin are placed 
on a cone or plane. Some configurations and excitations produce radiation 
patterns which are distinctly different from those obtained heretofore. 



* A convenient abbreviation for logarithmic spiral, a synonym for equiangular 
spiral. 

1. J. D. Dyson, "The Equiangular Spiral Antenna," IRE Trans, on Antennas and 
Propagation, vol. AP-7, pp. 181-187, April, 1959. Also Technical Report 
No. 21, University of Illinois Antenna Laboratory, September 15, 1957. 

2. J. D. Dyson, "The Unidirectional Equiangular Spiral Antenna," IRE Trans.. 

on Antennas and Propagation, Vol. AP-7, October 1959* Also Technical Report 
No. 33, University of Illinois, Antenna Laboratory, July 10, 1958. 



It is the purpose of this paper to introduce a simple theory relating 
the excitation and the radiation fields (insofar as now possible), and to 
present data showing the performance of the four-arm conical log-spiral 
antenna with one particular excitation. 



2. THE CONICAL LOG-SPIRAL ANTENNA 
2.1 The Basic Structure 

The conical log-spiral antenna with its associated coordinate system 
is shown in Figure 1. On a plane surface, the edges of one arm of a 

logarithmic spiral antenna may be defined by 

'■ ' i aff 

P = P e 

and 

/ a a(<P - $) 



Since Tan a = 



dPV.d0> . a 

/ / ( — - — ) <P 
P l =P e tan a 

The orthogonal projection of P and P_ on the surface of revolution 

1 ^ 

7T - 9 = G is defined by sin 9 

tan a ' r 

sin f 
a " d -• (r -2) (<P-h 

where • -sin 6 

2_ 6 . / 

tan a 
K = e 

P is the radius vector from the origin to the truncation of the spiral 
at the apex region, and P and p are the radius vectors to the inner and 
outer edges of the exponentially expanding arm at a given angle <P. The 
angle ° is a constant and of such value that if the curve traced out by 
P is rotated about the axis through the angle 6 it will coincide with the 
curve traced out by p . The angle a, a constant, is the angle between the 
radius vector and a tangent to the log-spiral curve at the point of inter- 
section. 




Figur* 1 A conical log-spiral antenna with 
associated coordinate system 



5 

The second arm of the balanced structure is defined by rotating curves 
1 and 2 through 7T radians. Hence the defining parameters are: the included 
cone angle, 26 ; the arm width determined by °, or K; the rate of spiral, a, 
the base diameter, D; and the apex diameter, d. 
2.2 The Radiation Pattern Beamwidth 

The beam width of the two-arm conical antenna can be controlled over 

a limited range by a suitable choice in the rate of spiral. Figure 2 

shows typical electric field radiation patterns for balanced two arm 

.o „„o , ,_o 



ant 



ennas constructed with an a of 73 60 and 45 . Typical half-power 



beamwidths range from 60-70° for a of 82 , 70-80 for a of 73°, and 160-180° 

o o 

for an a of 60 . As the angle a is decreased to 45 the beamwidth increases 

o 
to 180-200 . Pattern cuts through the axis and perpendicular to the axis 

of the antenna [Figure 2(c)] indicate that this latter case provides essentially 

circularly polarized coverage in one hemisphere and omnidirectional coverage 

on the 6 = 90 plane. 

It has been pointed out that a modified version of the balanced conical 

antenna is obtained when the width of the arm is made constant rather than 

2 

tapered . This form of the antenna is readily constructed of wire or cable. 

However, it is only an approximation to the true equiangular spiral structure. 
The approximation is good for relatively tightly spiraled antennas, i.e., a 
greater than 60 , and results in only minor pattern changes. As the angle 
a is decreased to the neighborhood of 45 to 50 marked pattern changes occur 
for the wire approximation, including a multilobing of the main beam and 
large radiation off the base of the cone. Thus to realize the patterns such 
as shown in Figure 2(c) the decrease in the rate of spiral must be accompanied 

by an increase in the arm width, i.e. increase in o (or K) . The patterns shown 

o 
in Figure 2(c) were for an antenna constructed with the parameters; 29 = 20 , 

o 

a = 45° K = .75 ( 6 = 94°). 



<f> =0°, 9 VAR 




(1=73' 



<£VAR, e=90 < 




^.=60* 



b. 



«=45 ( 





Figure 2 Variation in electric field pattern 
of typical balanced 2 arm conical 
equiangular spiral antenna 

E^, Eg polarization. 



9 = 10 . 
o 



3. THE CONICAL BEAM ANTENNA 



3.1 The Principle 

When using multiple-arm structures the number of choices of feeding 

3 
systems increases . There are basic excitations of multiterminal antennas 

which are simply related to the azimuthal variations of fields of the 

- -jirfp 
form e J associated with solutions of Maxwell's equations. The parameter 

m must be an integer to make the field single-valued. 

Excitations of the spiral arms which correspond to each of these 

radiation "modes" are readily apparent. The customary excitation of the 



two-arm spirals, as shown in Figure 3(a), corresponds to m = + 1 and would 
be expected to produce a field which varies primarily as e . For antennas 
which are not large compared to the wavelength the lower order terms will 
be predominant. 

With four-arm structures the number of basic excitations increases. 
An excitation corresponding to m = 1 is shown in Figure 3(b) . Hence in 
order to obtain the e fields with a four-arm spiral it is necessary to 
excite the two pairs of arms with a 90 degree phase shift between them. 

With the four-arm structures it is possible to produce fields which 
correspond to higher values of m . For example, the excitation shown in 

Figure 3(c) corresponds to m = +2 and should produce fields which vary 

-i2fP 
primarily as e . This concept may be generalized to any number of arms, 

N, and a discussion of the more general case is given in the Appendix. 



3. G. A. Deschamps, "impedance Properties of Complementary Multiterminal 

Planar Structures," Trans. IRE, Special Supplement, Vol. AP-7, Dec, 1959, 
p. S371. Also Technical Report No. 43, University of Illinois, Antenna 
Laboratory, Nov. 11, 1959. 

* First pointed out in Quarterly Report No. 5, Contract AF33(616)-6079, 
Antenna Laboratory, University of Illinois, 31 December 1959, pp. 9-11. 



+ CT7 (a) 







+j 

-I + IC=3 (b) 

"J 



fl 

D 






fl 



Figure 3 Possible feeding arrangements 
for multi-arm structures 



9 

Examination of the solutions of Maxwell's equations in spherical 
coordinates shows that values of m different from unity are always accompanied 
by a null in the 9 - function (associated Legendre polynomials) along the 
polar or 8 = axis. Therefore we expect an excitation which corresponds 
to any in / + 1 to produce a conical beam. The excitation of four arms 
corresponding to m = 2 as shown in Figure 3(c) , is the simplest case. 

This lowest order conical beam excitation is readily achieved by 
connecting opposite arms together and feeding one pair against the other, 
i.e. 180 degrees out of phase. It is apparent also from the symmetry of 
the input currents in this case that there will be zero field along the 
antenna axis. The antenna can be fed by a balanced feed line, or a coaxial 
line and balun, placed on the axis of symmetry. It may also be fed by 

carrying the feed cable along one of the arms as outlined in the previous 

2 
paper m Details of this latter method are shown in Figure 4. The balance 

and symmetry of the feed is important if symmetrical patterns are desired. 

3.2 Radiation Patterns 

Figure 5 shows typical radiation patterns of symmetrical four arm 

antennas fed in the manner shown in Figure 3(c). As indicated, the rate 

of spiral, (the parameter a), which was the primary factor in controlling 

the beamwidth of the balanced two arm antenna, determines the orientation 

of the conical beam of the balanced four arm antenna. Conical antennas may 

be constructed to provide a conical beam with any angle of orientation 

from around 40 to more than 90 off the axis of the antenna. The case 

where the beam maximum is located at 9 = 90 is of particular interest 

since it fills a need for a simple, very broad band, circularly polarized, 

omnidirectional source. 



10 



\ 



\ 



TOP 






\ 



V 



SIDE 




Figure 4 'infinite balun" feed used 

on a four arm conical 

beam antenna 




<*=73° 



oC = 60° 



11 




Q^=45° 



uj 

UJ 

o 



CD 



40 r 



50 - 



60 



70 - 



- 80 



90 



/' 



•J- 



40 50 60 70 

o< IN DEGREES 



80 90 



Figure 5 Typical electric field patterns and 
orientation of the conical beam as a 

function of the rate of spiral (7.5 ^ < 10 ) 



• ■ 



. 



Figure 6 shows a typical four arm balanced equiangular spiral antenna 
constructed on a 15 cone. This antenna was etched from a flexible, copper- 
clad, teflon-impregnated, fiberglass material and then formed into a cone. 
The feed cable is rg 141/U. The energized cable is carried along one arm; 
dummy cables are placed on the other arms to maintain structural symmetry. 
To obtain the desired bandwidth, the arms on this particular antenna were 
later extended to a cone base diameter of 31 centimeters. Radiation 
patterns of this antenna are shown in Figures 7 and 8 from 550 mc where the 
base is .57 wavelengths in diameter up to 4000 mc where the diameter of 
truncated apex is approximately 0.2 wavelengths. The patterns are for E~ 
and E.~ polarized fields. The first two columns are pattern cuts through 
the axis of the antenna and the third column is for a cut perpendicular to 
the axis, on the U = 90 plane. 

The azimuthal coverage shown in these patterns may be examined in greater 
detail in Figure 9 where the total deviation in decibels from omnidirectional 
coverage is plotted for the orthogonal polarizations. The axial ratio 
on the 6 = 90 plane varies somewhat with the angle i P. It is shown for 
one particular angle, which is a representative angle of orientation and 
not an optimum case. Over a considerable bandwidth the total amplitude 
deviation is less than 3 db and the axial ratio is 3 db or less. 

Two pattern characteristics should be noted. For large a, the conical 
beam patterns are smooth and well formed and, if desired, the arms may be 
approximated by wire or cable. As the angle a decreases beyond 60° the 

on patterns exhibit minor irregularities and are not as symmetrical. 
In ;i ■« loosely spiraled antennas require wider exponentially 

expanding arms. Th< ••-.<■ ' h. istics correspond to those noted for the 




13 



Figure 6 Antenna C - 15 - 9 etched from copper clad 
teflon impregnated fiberglass 
a = 45° D ■ 20.5 cm 



14 



8°, OVcuu. 
F=550Mc 




700 Mc 



CJ>-90° e vcuo. 





(j)Vojo Q = 9Q < 





900 Mc 






N \ 




S \ 


/ 
r 




^ 









1200 Mc 




Figure 7 Electric field patterns of a 
balanced symmetrical 4 arm conical 
ilangular spiral antenna. 

.", a 1 .", K = .925, D = 31 cm, d = 1.5 cm 




0s 0° e "z^t/. 
1600 Mc 



^=90° e vcuu. 



(£ VoMj. ^ Q r 9QO 15 




3000 Mc 






^ 



"h 



'y.c. .^'^^s-:^ 



V-< ' -v 



7 ^?r 




4000 Mc 






y 






s; 






^r=— 









\ 

n 



I : 



\ ' 



/ 
I / 



-^~ 



e = 



Figure 8 Electric field patterns of a 

balanced symmetrical 4 arm conical 

equiangular spiral antenna. 

7.5°, a = 45°, K = .925, D = 31 cm, d = 1. 5 cm 





a 

o 

•H 
+-> 

cS 

•H 

•o 

<D 

X! 00 
-P 

eg 
«H 

O «> 

ID (0 
bD <D 

Q) U> 
> 'H 

O Ek 

o 
c 

•H 
(0 

c 

Fh 

0) 

-p 
+-> 
a 
a 



X! 
■p 

S 
•H 
N 



a) 
h 

•H 



o08l = 4> 

o06=e 

OllVti 1VIXV 



qp Nl NfcGlJMd 

Nonviavd o06=e jo 

N0I1VIA3Q 3anilldkNV 



17 
two arm axial beam antennas. 

3.3 Pattern Beamwidth 

The beamwidth in a <P = constant plane is relatively insensitive to a 
change in antenna parameters. Antennas constructed with both 15 and 20 

included cone angles (2« ) with 45 < a < 73 and with cable arms or with 

o - - 

exponentially expanding arms had half-power beamwidths ranging from 35 to 55 
degrees, with an average value of 45 degrees. 

3.4 The Input Impedance 

The input impedance of the four arm antenna, fed in the manner of 

Figure 3(c), rapidly converges to a characteristic value. Antennas constructed 

o o 
with 15 or 20 included cone angles with RG 8/U arms or exponentially 

expanding arms fed with RG 141/U, typically have an input impedance of 

from 45 to 55 ohms for a ranging from 45 to 60 degrees. As a is increased 

to 73 degrees the impedance rises to the neighborhood of 70 ohms. These 

values are approximately one half those noted for similar two arm antennas. 

The input voltage standing wave ratio of the antenna referred to in 

Figures 7 and 8 is plotted in Figure 10. Note that it is less than 1.5 

to 1 referred to 50 ohms over most of the usable pattern bandwidth. 

3.5 Operating Bandwidth as a Function of Antenna Size 

The usable antenna bandwidth is fundamentally determined by the diameter 
of the truncated apex and the antenna arm length. As with the two arm 
antennas, the radiation patterns tend to deteriorate as the apex region 
approaches 1/4 wavelength in diameter. It was previously noted that the 
balanced two arm antenna constructed on a 15 or 20 degree (total apex angle) 
cone, with an a of 73 could be operated to a frequency such that the cone 
base diameter is on the order of 1/3 wavelength. As the rate of spiral 







ed 
U 

•H 
C 

o 
o 

E 
u 

a 
•<# u 

•H 

h a 

O 

•H U 

a eS 

>> rH 

tuD 

<H C 

rt 

•H 

^ cr 
to <D 
> 



s-t 

bO 






« 



e 

o 



ro 






CD 



in 

<* 

ll 



U~ OS 01 Q3dd3J3d tiMSA 



19 



is decreased, i.e. a decreased to 45 there is not sufficient radiation 
surface on this size cone to dissipate the energy without back radiation 
and hence the size of the cone must be increased to the order of .6 
wavelength at the lowest frequency of operation. The four arm structures 
exhibit very similar characteristics and hence omnidirectional coverage on 
the = 90 plane requires an antenna whose base diameter is on the order 
of .6 to 2/3 wavelength at the lowest operating frequency. 



20 

4. A NON-FREQUENCY INDEPENDENT VERSION 

Thus far we have considered only antennas constructed from the 
equiangular spiral curve. These antennas are frequency independent 

in the sense that, within the limits imposed by the physical size, the 

4 
scaling principle is fulfilled . The pattern characteristics of these 

log-spiral antennas (such as the beamwidth of the two arm antennas and the 

angle of orientation of the conical beam of the four arm antennas) are 

constant for a change in the frequency of operation. These characteristics 

are directly related to the constant parameter a, which indicates the rate 

of spiraling of the arms. 

It is possible, as shown in Figure 11, to construct conical antennas 

from other curves, such as the Archimedes spiral. Although these antennas 

may be operated over wide frequency bands, they are not frequency independent 

since the parameter a at any point on the curve is directly related to the 

angle <P at that point. As the frequency of operation is changed the active 

aperture of the antenna is composed of a structure with a changing rate 

of spiral. This shows up as a definite widening of the beamwidth of the 

two arm conical Archimedes spiral antenna as the operating frequency is 

increased. There is also a variation in the angle of orientation of the 

conical beam o! the four arm conical Archimedes spiral antenna with a 

change in frequency. 



A. V. H. Kurmey, Frequency Independent Antennas," 1957, IRE National 

"1, P'. 1, pp. 114-118. Also Technical Report No. 20, 
University '-f tlllnoie, Antenna Laboratory, October 25, 1957. 



p =ke 



Q(j> 



21 




a = arctan tt 



(a) 



p = ka<£ 




<< = arctan <£ 



(b) 

Figure 11 Projection of equiangular spiral 

and Archimedes spiral curves on 

a conical surface. 



22 

Radiation patterns for a four arm conical Archimedes spiral antenna 

are shown in Figure 12(a) . This antenna was constructed to provide a range 

o o 

of a from approximately 45 at the apex region to 85 at the base. As 

indicated in Figure 12 the complete range of beam orientation from 

o o 
approximately 45 to 90 off axis is swept out as the frequency is varied 

from 1000 to 2000 mc . For comparison, patterns for an equiangular spiral 

antenna wound on the same cone are shown in Figure 12(b) . 



ARCHIMEDES SPIRAL 



P =1.026* 




f = 1000 Mc 
E * 



B 



EQUIANGULAR SPIRAL 



( 

/>= e 



sin 10° 
tan45° 



)4> 




23 




f = 1400 Mc 





f = 2000 Mc 




Figure 12 Electric field patterns of 
symmetrical 4 arm conical 
antennas 

© = 10°, D = 29.5 cm, d = 4.5 cm 

o ' ' 

(<P = 90°, Q var pattern) 



24 
5. CONCLUSIONS 

It is possible to obtain a conical beam from the balanced equiangular 
spiral antenna by constructing the antenna with more than two arms and 
connecting these arms to provide a higher order <P - variation which is 
accompanied by an axial null. The angle of this conical beam can be 

controlled; in particular, it can be placed in the immediate 

o 

vicinity of the 6 = 90 plane to provide an omnidirectional pattern. 

An antenna with four symmetrically spaced arms can provide a radiation 
pattern that is within 3 db of omnidirectional coverage. The standing 
wave ratio of this antenna, referred to a 50 ohm coaxial cable, is usually 
less than 1.5 to 1 over the pattern bandwidth. 

This form of log-spiral retains the extremely wide frequency band- 
width and circular polarization properties of the basic conical log- 
spiral antenna and it provides a coverage which heretofore has been difficult 
to obtain even with narrow band antennas. 

Since the Archimedian spiral approximates a log-spiral with changing 
parameters, this non-frequency-independent version of the antenna may be 
constructed such that the conical beam may be frequency scanned in the 
polar angle. 



25 

APPENDIX 

Consider a radiating system composed of a number of identical conductors 

on the surface of a cone. Assume that the structure has N-fold rotational 

2ir 
symmetry, i.e., a rotation about the cone axis through the angle will 

leave the structure unchanged. Figures 3(b) and 3(c) of the text show 

examples of four-fold symmetry. 

The excitation of the conductors will be accomplished at the apex of 

the cone where the dimensions are small compared to the wavelength. There 

will be N terminals available in the excitation region, symmetrically 

spaced about a circle of small radius. Denoting the input current at the 

n terminal by I , the excitation can be described by the current vector 



I = ( V V v v w 

Any index is defined modulo N. To satisfy the requirement of conservation 
of current, we note that 



N 

£ 1=0 (1A) 

T n 

n = 1 



This viewpoint that each possible excitation is represented by a 
vector, I leads to the examination of the possible basis for the vectors 
of this space. Because of the n-fold symmetry the choice of "symmetrical 
components" as base vectors proves more convenient than others. 

Rotation of the excitation by one step 

: a = ( v v v V (2A) 



26 



® 



/ 



© 



© 



© 



© 



Figure 1A Terminal region of a 
structure having N - fold 
rotational symmetry 



27 



2 IT 

would simply produce a field which is rotated in space by the angle " 
Hence the transformation of excitation 

I 1 as PI (3A) 

where P is th*~ n x n permutation matrix 

1 ... 

1 ... 

P= 0001. ..00 (4A) 
1 

1 

produces a simple change in the field. The matrix P is a special case 

. 27Tk x 
of circulant matrices. The eigenvalues of P are exp. ( — ) and the 

eigenvectors are 

27TR . 27Tk(N-2) . 27Tk(N-l) 

-1/2. J N~ J N J N . 

A k = N (1, e , , e , e ) 



(5A) 

From these eigenvectors we can obtain the basis of our vector space. 
The eigenvectors of P for the case N = 4 are 
A 1 = 1/2 (1, j, -1, -j) 



A 2 = 1/2 (1, -1 ; 1, -1) 
A 3 = 1/2 (1, -j- -1, j) 



(6A) 



A 4 = 1/2 (1, 1, 1, 1) 

Note that A A and A satisfy the condition of Equation (1A) whereas 
A does not. The former three vectors provide an orthonormal basis which 
spans the vector space of all possible excitations when N = 4. 



28 

A . A * = 1/4 (1 + 1 + 1 + 1) = 1 

A 2 ' A 2 = 1/4 (1 + 1 + 1 + 1) = 1 

A 3 . A * = 1/4 (1 + 1 + 1 + 1) = 1 

A . A * = 1/4 (1 - j - 1 + j) « (7A) 

A . A * = 1/4 (1 - 1 + 1 - 1) = 

A 2 . A 3 * = 1/4 (1 - j - 1 + j) = 

Let us now examine the properties of these basic excitations. 
Excitation I = A produces field in which a rotation by 77/2 is equivalent 
to a 7T/2 change in phase. Solutions of Maxwell's equations in spherical 
coordinates can be written with azimuthal variation of the form e (m integer) 
Of these solutions only those with m = 4k + 1 (k = 0, + 1, + 2, +3...) 
will satisfy the above relationship between rotation and change in phase. 
Hence the excitation I = A will produce fields which can be expressed 
in the following way 



F, = L a (P 

k 4k + 1 (8A) 



where <Pm = g (r 9) e J 

in ^ 



nfP 



Similarly, I = A yields m = 4 k + 2 and I = A yields m = 4 k + 3, 
^ 3 

and these each will produce fields 

F = E a <P 
P k 4k + p (9A) 

A Bimil.ir ftrgtUMnl Cftl i <«d through for arbitrary N. 

Ari satisfying (1A) can be expressed as a linear combination 

A ] " A 2' A 3 '' Produced thereby as a linear combination of 

F ji y y T, " 15i '><>>> de«( ribed by E will produce the fields 



29 



F = E (A * E*) F (10A) 

J J 

If the coefficients a above were known, the fields produced by any 
excitation would be completely determined. Unf ortunately, this problem 
has not yet been solved for log-spiral elements. 

Some useful observations can be made for the log-spiral however, 
by interpreting the above results in terms of experimental data. 
Consider the case N = 2 which has been extensively investigated. The 
eigenvectors for the permutation matrix in this case are 



\ " \ (1 > ^ 



A 2 = 1(1,D 



(11A) 



and only A satisfies Equation (1A) . This corresponds to the excitation 
of the two-arm spiral sketched in Figure 3(a) . The field F for the case 
N = 2 would be of the form 



F = £ a (P 
Ik T 2k + 1 (12A) 

and the possible ^-variations are 

e j(2k + 1) 9 
It has been observed that the radiation produced by 2-arm log-spiral 
antennas is very nearly circularly-polarized over the major portion of 
the beam. Circular polarization with non-zero field on the axis requires 
that E^ and Eg vary as cos(<»>t + <f) and sin (<*>t + <P) respectively, the m = 1 
case. Functions with higher values of m would contribute to the off-axis 
fields. Figure 2A shows a sketch of how the circularly polarized fields 



= 0, 





m=2 




m = 3 



7.h Instantaneous electric vectors at 
■ and ■ for three values of m. 



31 
would appear instantaneously around the polar axis for a few of the small 
values of m. The values m = 1 and 3 would predominate for the two-arm 
case. Note that these add alternately in and out of phase in orthogonal 
cross-sections. Hence the rotational symmetry of the beam should provide 
an indication of the relative magnitude of these two fields, neglecting all 
others. A tightly- wound spiral produces more nearly rotationally-symmetric 
beams, hence predominantly the m = 1 case. 

The excitation of Figure 3(c) used to produce the conical beams is one 
of the eigenvectors of (6A), namely A . This excitation produces fields 

F = £ a, <P 

2 k 4k + 2 

and the experimental results indicate that, for certain parameter choices, 
the lowest order terms of this series predominate 

F ~ a <P n + a <P „ 
2 ~ o 2 -1-2 

These two cases seem equally likely on the basis of the order of the 

functions involved. However these two cases correspond to circular 

polarized fields of opposite sense. One sense of polarization is favored 

over the other by the direction of the spiral winding. Hence it is reasonable 

that one of the coefficients a . a be small, a conjecture which is again 

o' -1 

confirmed experimentally.